Problem Proposed for the American Mathematical Monthly

Journal Article · Fri Nov 14 00:00:00 EST 2008 · American Mathematical Monthly

OSTI ID:963543

Bailey, David H; Borwein, Jonathan M; Waldvogel, Jorg

The problem is to define: P(x) := {Sigma}{sub k = 1}{sup {infinity}} arctan (x - 1/(k + x + 1) {radical}(k + 1) + (k + 2) {radical}(k + x)). (1) (a) Find explicit, finite-expression evaluations of P(n) for all integers n {ge} 0. (b) Show {tau} := lim{sub x {yields} -1{sup +}} P(x) exists, and find an explicit evaluation for {tau}. (c) Are there a more general closed forms for P, say at half-integers? Solution with the abbreviations: r := {radical} (k + 1), s := {radical} (k + x) the argument of arctan in (1) becomes s{sup 2} - r{sup 2}/(s{sup 2} + 1) r + (r{sup 2} + 1) s = s - r/r s + 1 = 1/r - 1/s / 1 + 1/r 1/s.

• Research Organization: Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
• Sponsoring Organization: Computational Research Division
• DOE Contract Number: DE-AC02-05CH11231
• OSTI ID: 963543
• Report Number(s): LBNL-2137E; TRN: US200918%%328
• Journal Information: American Mathematical Monthly, Journal Name: American Mathematical Monthly
• Country of Publication: United States
• Language: English

Related Subjects

97
EVALUATION
ANALYTIC FUNCTIONS
MATHEMATICAL SOLUTIONS